.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/foundations/plot_05_information_rates.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_foundations_plot_05_information_rates.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_foundations_plot_05_information_rates.py:


Information rates of QAM
========================

The mutual information :math:`I(X; Y)` of a constellation over AWGN is the rate a
receiver that decodes symbols can achieve; the generalised mutual information (GMI)
is the rate of a bit-interleaved receiver that decodes bits independently, the usual
coded-modulation receiver. Both stay below the Shannon capacity
:math:`\log_2(1 + \mathrm{SNR})` and saturate at :math:`\log_2 M`; with Gray labels
the gap between them is small.

The curves are computed by Gauss–Hermite quadrature (:func:`comnum.theory_mi`,
:func:`comnum.theory_gmi`); the markers are Monte Carlo estimates on 100 000 noisy
16-QAM symbols (:func:`comnum.information_rates`).

.. GENERATED FROM PYTHON SOURCE LINES 16-57

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import comnum
    from comnum.plot import BACKGROUND, GREY, ORANGE

    snr = np.arange(-5.0, 30.5, 1.0)
    fig, ax = plt.subplots()
    ax.plot(snr, np.log2(1 + comnum.db2lin(snr)), color=GREY, linewidth=1.4, label="Shannon capacity")
    for m in [4, 16, 64]:
        mi = [comnum.theory_mi(m, s, "qam") for s in snr]
        gmi = [comnum.theory_gmi(m, s, "qam") for s in snr]
        (line,) = ax.plot(snr, mi, label=f"{m}-QAM, MI")
        ax.plot(snr, gmi, color=line.get_color(), linestyle="--", linewidth=1.4, label=f"{m}-QAM, GMI")

    # Monte Carlo estimates for 16-QAM
    c = comnum.Constellation("qam", 16).normalized()
    est = []
    for i, s in enumerate(snr[::3]):
        rng = comnum.Rng.stream(11, "mi", i, 0)
        idx, x = comnum.symbols(c, 100_000, rng)
        est.append(comnum.information_rates(comnum.awgn(x, s, rng), idx, c))
    ax.plot(
        snr[::3],
        [e["mi"] for e in est],
        "o",
        color=ORANGE,
        markeredgecolor=BACKGROUND,
        label="16-QAM, MI simulated",
    )
    ax.set(
        xlabel="SNR $E_s/N_0$ (dB)",
        ylabel="bits per symbol",
        xlim=(-5, 30),
        ylim=(0, 7),
        title="Information rates over AWGN",
    )
    ax.legend(ncol=2, loc="lower right")
    fig.tight_layout()




.. image-sg:: /auto_examples/foundations/images/sphx_glr_plot_05_information_rates_001.png
   :alt: Information rates over AWGN
   :srcset: /auto_examples/foundations/images/sphx_glr_plot_05_information_rates_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 58-61

The gap to capacity at 3.5 bits per symbol, which needs an SNR of
10 log10(2^3.5 - 1) = 10.13 dB: 16-QAM, close to saturation, is 1.4 dB away; 64-QAM
0.7 dB (what remains is the shaping gap of uniform square constellations).

.. GENERATED FROM PYTHON SOURCE LINES 61-70

.. code-block:: Python

    target = 3.5
    for m in [16, 64]:
        lo, hi = 0.0, 30.0
        for _ in range(50):
            mid = (lo + hi) / 2
            lo, hi = (lo, mid) if comnum.theory_mi(m, mid, "qam") > target else (mid, hi)
        print(
            f"{m}-QAM reaches {target} bits/symbol at {lo:.2f} dB (capacity: {comnum.lin2db(2**target - 1):.2f} dB)"
        )




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    16-QAM reaches 3.5 bits/symbol at 11.57 dB (capacity: 10.13 dB)
    64-QAM reaches 3.5 bits/symbol at 10.84 dB (capacity: 10.13 dB)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 14.690 seconds)


.. _sphx_glr_download_auto_examples_foundations_plot_05_information_rates.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_05_information_rates.ipynb <plot_05_information_rates.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_05_information_rates.py <plot_05_information_rates.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_05_information_rates.zip <plot_05_information_rates.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_